Nuclear fission reactions produce nuclear waste but nuclear fusion
reactions dont. Is it possible to do a nuclear fusion reaction with the
waste products of a fission reaction and thus eliminate nuclear waste?

Al Ghiorso, who smashes heavy atoms together with large accelerators
at Berkeley in California, and who discovered element 104 and named it
Rutherfordium after Ernest Rutherford, responded.

In fission the nucleus of a heavy atom such as uranium is split into two
lighter nuclei with a release of energy. In fusion two light atoms such as
hydrogen are pushed together to form one heavier nucleus, again with the
release of energy. Fission reactors have been in commercial use since the
1950's. Fusion holds great promise but because of the difficulty of pushing
two nuclei close together, power production has not yet been achieved other
than in the unusual environment of a hydrogen bomb which uses a fission
bomb as a detonator. Special high powered lasers are being developed to
hopefully produce fusion power reactors.

Fission reactions leave behind highly radioactive nuclei, some of which
have lifetimes of thousands of years. Can nuclear reactions be performed
to convert these into non-radioactive nuclei? In general one cannot put
the pieces back together again.

If one bombards the waste fission products with nuclei from an accelerator
or a neutron reactor, in principle one can
convert them into atoms that are more stable and thus more long lived. They
would be 'burned' into other nearby elements with the end result being less
radioactivity. At least that is the theory but there is a real question as
to whether this could be cost effective. I doubt it. There was a Los Alamos
project aimed at seeing whether the idea was feasible. It was pursued for a
while about a decade ago.